As we are accustomed in all cases to refer direction to the horizontal and vertical lines, and as the meeting of these lines makes the right angle, it naturally constitutes the fundamental angle, by the harmonic division of which a system of proportion may be established, and the theory of symmetrical beauty, like that of music, rendered susceptible of exact reasoning.
Let therefore the right angle be the fundamental angle, and let it be divided upon the quadrant of a circle into the harmonic parts already explained, thus:—
| Right Angle. | Supertonic Angles. | Mediant Angles. | Subdominant Angles. | Dominant Angles. | Submediant Angles. | Subtonic Angles. | Semi-subtonic Angles. | Tonic Angles. | |
|---|---|---|---|---|---|---|---|---|---|
| I. | (1) | (⁸⁄₉) | (⁴⁄₅) | (³⁄₄) | (²⁄₃) | (³⁄₅) | (⁴⁄₇) | (⁸⁄₁₅) | (¹⁄₂) |
| II. | (¹⁄₂) | (⁴⁄₉) | (²⁄₅) | (³⁄₈) | (¹⁄₃) | (³⁄₁₀) | (²⁄₇) | (⁴⁄₁₅) | (¹⁄₄) |
| III. | (¹⁄₄) | (²⁄₉) | (¹⁄₅) | (³⁄₁₆) | (¹⁄₆) | (³⁄₂₀) | (¹⁄₇) | (²⁄₁₅) | (¹⁄₈) |
| IV. | (¹⁄₈) | (¹⁄₉) | (¹⁄₁₀) | (³⁄₃₂) | (¹⁄₁₂) | (³⁄₄₀) | (¹⁄₁₄) | (¹⁄₁₅) | (¹⁄₁₆) |
In order that the analogy may be kept in view, I have given to the parts of each of these four scales the appropriate nomenclature of the notes which form the diatonic scale in music.
When a right angled triangle is constructed so that its two smallest angles are equal, I term it simply the triangle of (¹⁄₂), because the smaller angles are each one-half of the right angle. But when the two angles are unequal, the triangle may be named after the smallest. For instance, when the smaller angle, which we shall here suppose to be one-third of the right angle, is made with the vertical line, the triangle may be called the vertical scalene triangle of (¹⁄₃); and when made with the horizontal line, the horizontal scalene triangle of (¹⁄₃). As every rectangle is made up of two of these right angled triangles, the same terminology may also be applied to these figures. Thus, the equilateral rectangle or perfect square is simply the rectangle of (¹⁄₂), being composed of two similar right angled triangles of (¹⁄₂); and when two vertical scalene triangles of (¹⁄₃), and of similar dimensions, are united by their hypothenuses, they form the vertical rectangle of (¹⁄₃), and in like manner the horizontal triangles of (¹⁄₃) similarly united would form the horizontal rectangle of (¹⁄₃). As the isosceles triangle is in like manner composed of two right angled scalene triangles joined by one of their sides, the same terminology may be applied to every variety of that figure. All the angles of the first of the above scales, except that of (¹⁄₂), give rectangles whose longest sides are in the horizontal line, while the other three give rectangles whose longest sides are in the vertical line. I have illustrated in [Plate I.] the manner in which this harmonic law acts upon these elementary rectilinear figures by constructing a series agreeably to the angles of scales II., III., IV. Throughout this series a b c is the primary scalene triangle, of which the rectangle a b c e is composed; d c e the vertical isosceles triangle; and when the plate is turned, d e a the horizontal isosceles triangle, both of which are composed of the same primary scalene triangle.
Thus the most simple elements of symmetry in rectilinear forms are the three following figures:—
- The equilateral rectangle or perfect square,
- The oblong rectangle, and
- The isosceles triangle.
It has been shewn that in harmonic combinations of musical sounds, the æsthetic feeling produced by their agreement depends upon the relation they bear to each other with reference to the number of pulsations produced in a given time by the fundamental note of the scale to which they belong; and that the more simply they relate to each other in this way the more perfect the harmony, as in the common chord of the first scale, the relations of whose parts are in the simple ratios of 2:1, 3:2, and 5:4. It is equally consistent with this law, that when applied to form in the composition of an assortment of figures of any kind, their respective proportions should bear a very simple ratio to each other in order that a definite and pleasing harmony may be produced amongst the various parts. Now, this is as effectually done by forming them upon the harmonic divisions of the right angle as musical harmony is produced by sounds resulting from harmonic divisions of a vibratory body.
Having in previous works[7] given the requisite illustrations of this fact in full detail, I shall here confine myself to the most simple kind, taking for my first example one of the finest specimens of classical architecture in the world—the front portico of the Parthenon of Athens.